\beginsection{1.4}

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(a) Guess a formula for $1+3+\cdots+(2n-1)$ by evaluating the sum for
$n=1$, 2, 3, and 4. [For $n=1$, the sum is simply 1.]

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For each $n$ we have
$$\eqalign{
1&=1\cr
1+3&=4\cr
1+3+5&=9\cr
1+3+5+7&=16\cr
}$$
therefore a reasonable guess would be
$$1+3+\cdots+(2n-1)=n^2$$

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(b) Prove your formula using mathematical induction.

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We already have $n^2=1$ for $n=1$.
For $n+1$ we have
$$1+3+\cdots+(2n-1)+(2(n+1)-1)=n^2+2n+1=(n+1)^2$$
Therefore the formula is true for $n+1$ whenever it is true for $n$.
Hence by induction the formula is true for all $n\ge1$.

